Mathematics, mathematics education and computer programming.

Knowledge is Power

So, after a week away I have an opinion piece… Please give me feedback as I would like to hear other people’s opinions and I am open to mine being changed!

With the new GCSE specification coming in, I often hear how “we need to be teaching more problem solving skills”, “developing independent learners”, “using inquiry based teaching techniques”. This, along with a general focus on child centered learning makes me feel a bit uneasy….

I’m not against these techniques – I use investigations and inquiry type things to promote pupils curiosity about mathematics and give them a flavour of what being a mathematician is about – but I think they need to be used in conjunction with more traditional teaching styles to be effective.

Most of the maths taught in schools is hundreds of years old and required hundreds of years of work to be discovered and perfected. The average child isn’t going to be like Gauss. When asked to add up the first 100 numbers I’m sure most will just add them in turn (I may try this with some of my classes), even if some notice that you can work inwards and pair up the numbers to quickly work out sum with a multiplication, I don’t think even my sixth formers would come up with the algebraic formula for the sum of the integers without any prompts! There are some things which probably just need to be learnt, so that they can be applied. After all, mathematics is advanced by people applying previous knowledge in novel situations.

With some of the more extreme types of independent learning I question the benefit that it brings to the pupils. If they have no clue about something that has been presented to them, then even the most tenacious pupil will become demoralised. I can’t imagine a teacher that wouldn’t provide more guidance in this situation, and steer the class to the desired conclusion. I really like the unpredictability of the inquiry idea and being able to look at some maths that the class stumbles on, but for them to be able to do this they need a sound knowledge base. This is why, for students to be successful at this and other investigations time must be spent building up a good body of knowledge, that all pupils can successfully weild to enable them to make progress.

In terms of problem solving skills, I think these are probably best taught by equipping students with enough knowledge to feel comfortable that when an approach doesn’t work they can try something else. Tenacity and patience I think are the most important problem solving skills.

I guess in conclusion I don’t like the polarised investigative child centered vs didactic teaching debate – there is a place for both approaches and the good teaches will blend the methods to ensure the best progress of their pupils. But I believe a strong knowledge base is key for any success in mathematics.

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10 thoughts on “Knowledge is Power”

Great post … I don’t think problem solving is a distinct skill as such – you can of course teach strategies to approach a problem (i.e. use x for the unknown, draw a picture etc) but without a body of knowledge that students can call on its a pointless exercise. For example a student may be able to communicate “I know I need to find the area of that side of the triangle” but not know how to do it .. so for me knowledge first, then alongside teach appropriate skills.
I don’t think my style of teaching will change – the major change for me will come from being able to call upon skills from a wider variety of topics when approaching a problem than we do currently (and that is why knowledge has to come first!) – not a massive change as there are already obvious “partners” such as forming equations with geometry properties but the more students are exposed to these types of “Mix and match” questions the better they will get at it. Part of the art of problem solving is trying to call upon past experiences of similar problems and solutions that were used.

Thanks for the very detailed comment. I’m looking forward to being able to call on more skills and make more links too, but as you say them knowing what to do and then being able to do it are very different things.

I think I’d agree with most of this, particularly your summary in the last paragraph. Some topics do lend themselves beautifully to a bit of discovery or inquiry – lots of geometry for example, and I think it can help pupils to build connections between topics if they have to do a bit of thinking for themselves. I’ve done a couple of really nice 3 act maths lessons such as the filing cabinet one for surface area of a cuboid, which do really require pupils to think and analyse what they are doing and why. However I think the structure has to be thought through carefully so that pupils don’t end up floundering around for ages and not making any progress – I don’t mean in the Ofsted measureable sense, I mean in terms of actually learning some new, useful mathematics.

The “discovery” model works well with things that can be deduced logically, and it’s nice to get pupils to do this, because it’s emulating how the mathematics was discovered originally. But sometimes this can’t work, and trying to shoehorn a fake problem or false context in just complicates things and doesn’t really add much.

I think the demoralisation thing is key – thinking and working mathematically are skills that need to be developed, and we shouldn’t expect pupils to be excellent at this straight away, particularly when working on a new concept as well.

As an aside, I have actually done the sum from 1-100 thing as an unrelated starter (most years on Gauss’s birthday); I’m always surprised that there’s at least one or two pupils who work out the shortcut, but this might be might be more to do with the fact that I sit there giggling while they spend five minutes adding the numbers from 1 – 20 and then start asking if this is all they are doing all lesson, than any indication of prodigal maths skills!

Cheers for the reply. Could you share more details of the filing cabinet lesson, I’m intrigued? Nice to hear about your students finding the shortcut, that’s great! I’m definitely going to try it. I think structure and demonstrating to the students that it’s ok to struggle are really important.

I am a discovery skeptic, in most (if not all) cases students need a starting point when it comes to studying Mathematics. However once prompted students in a number of topics, like Geometry, can be given opportunities to develop deeper understanding. The constructivist model has caused a lot of problems over the last 10 years, and we’re having to make up for that.

Agree there’s a risk of misinterpretation but I don’t think that the intention of ‘problem solving skills’ and ‘independent learner’ is to deliver investigative lessons, which are fairly well dismissed as rubbish in the Sutton Trust report. I think what they intend is for teachers to step away from linear questioning with no depth, and move towards problems that genuinely require thinking and analysis.
e.g. question 1 and question 7 here:http://solvemymaths.com/2015/03/31/taking-back-the-f-word/

I also think investigation has its place in lessons, although not as a ‘figure out this theorem if you can’ way, but more of a
‘can you find a square and rectangle with equal perimeter? Are their areas the same?’ kind of way.
The second example is approachable and is a lot less likely to stump students, and can be progressed easily (e.g. can you find a triangle and a … etc).
The first example causes massive headaches.
When do you ‘reveal?’
What do you do with the student who ‘gets it’ first?
What was the purpose of the whole exercise for the ones who needed the reveal anyway?
Are you genuinely aiding their understanding through this exercise?
How long until the task dwindles? Usually its different times for each student – which is a teacher nightmare.
How many students genuinely hide behind others because they just don’t know what to do?
I do use these kinds of lessons occasionally despite all these flaws, but they tend to work much better if you as the teacher don’t have a specific thing you want them to find out, and don’t base a whole lesson on it perhaps.
e.g. I asked students to ‘see if you could find something interesting’ about adding consecutive numbers together after challenging them to rewrite every number from 1-20 as an addition of consecutive numbers (post below)http://solvemymaths.com/2014/10/09/become-the-enabler/

The consecutive numbers post is lovely. I noticed similar things when I did that task earlier in the year. I love the problems on nRich! Your area and perimeter questions are great too, thanks for sharing them.

Absolutely loved this post! The biggest problem I face when teaching functional skills maths to adult learners is that they do not know how to apply their (limited) maths skills to a given problem solving scenario. This results in frustration on the part of the student and usually a gentle in the right direction from me.

I strongly believe that lack of confidence in basic skills does not help this scenario. I often find that adults with poor numeracy skills have low literacy skills. As such, students are at a further disadvantage because they literally don’t understand what the question is asking. As an example, a task I used recently involved completing an order form for equipment for candle making. The student is question had not appreciated that each candle needed a holder due to the limitation of her reading skills. It’s not just the maths jargon that trips the students up. It’s the every day words that are very easy to overlook.

I guess our job, as teachers, is to help students break the problems down into chunks perhaps by visualising the problem. Language skills need to be constantly challenged for this to happen. This is a slow process for some. Then we can think about the maths we might use to solve the problem.

Some of my students need a lot of guidance and support before they are ready for independent learning. I guess this makes sense considering their lack of confidence from experience. These barriers to learning take time, effort and a lot of patience to remove.